It is well known that expansion and connectivity properties of a graph can be read from its spectrum. As such, graphs exhibiting a spectral gap are very good expanders. In this talk, we will investigate these notions in the context of random graphs and see their stability under sparsification. We will observe phenomena reminiscent of the Baik-Ben Arous-Péché phase transition, and discuss connections with the study of outliers in random matrix theory. Based on joint works with Konstantin Tikhomirov.